In [1]}, [2]} we study real valued functions defined on a one codimensional homogeneous subgroup of Carnot groups of step 2 (see Section 5.1, [3]}) which including the Heisenberg groups. We define an appropriate notion of intrinsic derivative in this setting and we extend Theorem 1.3 and Theorem 5.7 in [4]} proved in \(\mathbb {H}^n\) .
Precisely, we prove that the intrinsic graph of continuous map \(\phi \)  is a regular hypersurface if and only if \(\phi \)  and its intrinsic gradient can be uniformly approximated by 1 functions and \(\phi \)  is distributional solution of PDEs' system \(D^\phi \phi =w\) , with \(w\)  is a continuous map. Moreover, we also show that these assumptions are equivalent to the fact that \(\phi \)  is broad* solution of \(D^\phi \phi =w\)  (see Definition REF ) and it is little 1/2-Hölder continuous (see Proposition REF  (1)). In \(\mathbb {H}^1\)  the notion of broad* solution extends the classical notion of broad solution for Burger's equation through characteristic curves provided \(\phi \)  and \(w\)  are locally Lipschitz continuous. In our case \(\phi \)  and \(w\)  are supposed to be only continuous then the classical theory breaks down. On the other hand broad* solutions of the system \(D^\phi \phi =w\)  can be constructed with a continuous datum \(w\) .
